Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 457019, 11 pages
doi:10.1155/2012/457019
Research Article
Devaney Chaos and Distributional Chaos in the
Solution of Certain Partial Differential Equations
Xavier Barrachina and J. Alberto Conejero
Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, Edifici 8E,
46022 València, Spain
Correspondence should be addressed to J. Alberto Conejero, aconejero@upv.es
Received 3 June 2012; Revised 13 September 2012; Accepted 17 October 2012
Academic Editor: Elena Litsyn
Copyright q 2012 X. Barrachina and J. A. Conejero. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
The notion of distributional chaos has been recently added to the study of the linear dynamics of
operators and C0 -semigroups of operators. We will study this notion of chaos for some examples
of C0 -semigroups that are already known to be Devaney chaotic.
1. Introduction
During the last years, several notions have been introduced for describing the dynamical
behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in
the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing
properties, and frequent hypercyclicity, among others. These notions have been extended, as
far as possible, to the setting of C0 -semigroups of linear and continuous operators. The recent
monograph by Grosse-Erdmann and Peris Manguillot 1 is a good reference for researchers
interested in the study of linear dynamics. In particular, it contains a chapter dedicated to
analyze the dynamics of C0 -semigroups. See also 2, which contains additional information
on further topics in the area.
In the sequel, let X be an infinite-dimensional separable Banach space. A C0 -semigroup
T is a family of linear and continuous operators {Tt }t≥0 ⊂ LX such that T0 Id, Tt Ts Tts
for all t, s ≥ 0, and for all s ≥ 0, we have limt → s Tt Ts pointwise on X.
We say that a C0 -semigroup T is Devaney chaotic if it is transitive and it has a dense set
of periodic points. On the one hand, a C0 -semigroup T is transitive if for any pair of nonempty
open sets U, V ⊂ X there is some t > 0 such that Tt U ∩ V / ∅. In this setting, transitivity is
equivalent to the existence of some x ∈ X with dense orbit in X, that is, {Tt x, t ≥ 0} X, see for
instance 1, Th. 1.57. This phenomenon is usually known in operator theory as hypercyclicity,
2
Abstract and Applied Analysis
and such a vector x is said to be a hypercyclic vector for T. On the other hand, a vector x ∈ X
is said to be a periodic point for T if there is some t > 0 such that Tt x x.
Other definitions of chaos, such as the one introduced by Li-Yorke and the one of
distributional chaos introduced by Schweizer and Smı́tal, have been also considered. The
relationships between these two notions in the Banach space setting have been recently
studied in 3. We recall that a C0 -semigroup T is said to be Li-Yorke chaotic if there exists
an uncountable subset Γ ⊂ X, called the scrambled set, such that for every pair x, y ∈ Γ of
distinct points, we have that
lim infTt x − Tt y 0,
t→∞
lim supTt x − Tt y > 0.
t→∞
1.1
Clearly, every hypercyclic C0 -semigroup is Li-Yorke chaotic: we just have to fix a hypercyclic
vector x ∈ X and consider Γ : {λx; |λ| ≤ 1} as a scrambled set, as it is indicated in 4, page
84.
Distributional chaos is inspired by the notion of Li-Yorke chaos. In order to define it,
given a subset B ⊂ R0 , we define its upper density as DensB : lim supt → ∞ 1/tμB ∩0, t,
where μ stands for the Lebesgue measure on R0 .
Definition 1.1. A C0 -semigroup T in LX is distributionally chaotic if there are an uncountable
set S ⊂ X and δ > 0, so that for each ε > 0 and each pair x, y ∈ S of distinct points, we have
Dens
s ≥ 0 : Ts x − Ts y ≥ δ 1,
Dens
s ≥ 0 : Ts x − Ts y < ε 1.
1.2
The set S is called the scrambled set. If S is dense in X, then T is said to be densely
distributionally chaotic.
A vector x ∈ X is said to be distributionally irregular for T if for every δ > 0
Dens{s ≥ 0 : ||Ts x|| > δ} 1,
Dens{s ≥ 0 : ||Ts x|| < δ} 1.
1.3
Such vectors were considered in 5 so as to get a further insight into the phenomenon
of distributional chaos, showing the equivalence between a distributionally chaotic operator
and an operator having a distributionally irregular vector.
The first systematic approach to distributional chaos for linear operators was taken in
6, where this phenomenon was studied in detail for backward shift operators. Later, Peris
and Barrachina proved that for translation C0 -semigroups on weighted Lp spaces, 1 ≤ p < ∞,
Devaney chaos implies distributional chaos. However, the converse does not hold. They also
provide an example of a translation C0 -semigroup that is distributionally chaotic but it is
neither Devaney chaotic nor hypercyclic 7.
Hypercyclicity and Devaney chaos are hard to observe directly from the definition.
The Hypercyclicity Criterion, in any of its forms, and the Desch-Schappacher-Webb Criterion
have turn out to be powerful tools in order to verify these properties. Very recently, Albanese
et al. have stated a criterion in order to show that a C0 -semigroup is distributionally chaotic
and has a dense distributionally irregular manifold 8. Our goal is to study distributional
chaos for some C0 -semigroups that are already known to be Devaney chaotic. The dynamics
exhibited by these C0 -semigroups will motivate us to pose some open questions.
Abstract and Applied Analysis
3
2. Criteria to Determine Devaney Chaos and Distributional Chaos
The following statement of the Hypercyclicity Criterion for C0 -semigroups is inspired by its
version for operators in 9.
Theorem 2.1 Hypercyclicity Criterion for C0 -semigroups; see 10, Th. 2.1, 11, Crit. 3.1,
and 1, Th. 7.26. Let T be a C0 -semigroup in LX. If there are a sequence {tn }n ⊂ R with
limn → ∞ tn ∞, dense subsets Y, Z ⊂ X and maps Stn : Z → X, n ∈ N such that
i limn → ∞ Ttn y 0 for all y ∈ Y ,
ii limn → ∞ Stn z 0 for all z ∈ Z, and
iii limn → ∞ Ttn Stn z z for all z ∈ Z,
then T is hypercyclic.
Sometimes the Hypercyclicity Criterion is hard to be applied, and in fact, it only
provides one of the ingredients of Devaney chaos. Moreover, in many situations, we can have
the infinitesimal generator of a C0 -semigroup but we do not have the explicit representation
of its operators. This is quite common when we deal with the solution C0 -semigroups
associated to certain partial differential equations. Desch et al. gave a criterion which permits
us to state the Devaney chaos of a C0 -semigroup in terms of the abundance of eigenvectors
of the infinitesimal generator.
Theorem 2.2 Desch-Schappacher-Webb Criterion; see 12, 13. Let X be a complex separable
Banach space, and let T be a C0 -semigroup in LX with infinitesimal generator A, DA. Assume
that there exist an open connected subset U ⊂ C and a weak holomorphic function f : U → X, such
that
i U ∩ iR / ∅,
ii fλ ∈ kerλI − A for every λ ∈ U, and
iii for any x∗ ∈ X ∗ , if fλ, x∗ 0 for all λ ∈ U, then x∗ 0.
Then the semigroup T is chaotic.
For the case of distributional chaos, Albanese et al. obtained the following sufficient
condition, inspired by the result for the discrete case given by Bermúdez et al. in 4.
Theorem 2.3 Dense Distributionally Irregular Manifold Criterion; see 8, Cor. 2. Let T be a
C0 -semigroup in LX. Assume that there exist
i a dense subset X0 ⊂ X such that limt → ∞ Tt x 0 for each x ∈ X0 , and
ii a Lebesgue measurable subset B ⊂ R0 with DensB 1 satisfying either
or X is a complex Hilbert space and B Tt −2 dt < ∞.
B
Tt −1 dt < ∞,
Then T has a dense manifold whose nonzero vectors are distributionally irregular vectors. (When this
happens, one says that T has a dense distributionally irregular manifold).
Furthermore, they also proved that a C0 -semigroup T is distributionally chaotic if,
and only if, T has a distributionally irregular vector 8, Th. 3.4. Therefore, Theorem 2.3 can
be also understood as a criterion for distributional chaos. In the sequel, we will apply this
criterion several times in order to determine that certain C0 -semigroups are distributionally
chaotic.
4
Abstract and Applied Analysis
3. Distributionally Chaotic C0 -Semigroups
In this section, we consider several examples of C0 -semigroups that are already known to be
Devaney chaotic, and we will study when they exhibit distributional chaos. These examples
will be considered on the following spaces:
p
Lρ I, C
f ∈ MI, C : f p,ρ 1/p
p
fs ρs ds
<∞ ,
with 1 ≤ p < ∞, 3.1
I
where I is an interval on R and ρ a weight function. If ρx 1, then we will simply denote it
as Lp I, C, 1 ≤ p < ∞. The hypothesis on ρ may be different on each example.
In 14, Takeo considered the following first order abstract Cauchy problem on
Lp I, C, 1 ≤ p < ∞:
∂u
∂u
ζx
hxu,
∂t
∂x
u0, x fx,
3.2
x ∈ I,
where ζ and h are bounded continuous functions defined on I. This ordinary differential
equation has been used to model the dynamics of a population of cells under simultaneous
proliferation and maturation 15. When ζx is constant and equal to 1 and I R0 , the
solution C0 -semigroup {Tt }t≥0 of 3.2 is defined as
xt
Tt fx exp
hs ds fx t,
x
∀x, t ≥ 0, f ∈ Lp R0 , C .
3.3
Theorem
3.1. If hx is a real function and there is a measurable set B ⊂ R0 such that DensB 1
x
and B exp−p 0 hsdsdx < ∞, then the C0 -semigroup {Tt }t≥0 defined in 3.3 is distributionally
chaotic on Lp R0 , C, 1 ≤ p < ∞.
x
Proof. If we define ρx exp−p 0 hsds, then the operators of {Tt }t≥0 can be rewritten as
Tt fx ρx
ρx t
1/p
fx t.
3.4
This function ρx is an admissible weight function in the sense of 12, Def. 4.1, which
ensures that the left translation semigroup {τt }t≥0 defined as
τt fx fx t,
p
is a C0 -semigroup on Lρ R0 , C.
p
for x, t ≥ 0, f ∈ Lρ R0 , C
3.5
Abstract and Applied Analysis
5
Let us define φfx ρx1/p fx and consider the following commutative
diagram:
p
Lρ (R+0 , C)
τt
p
Lρ (R+0 , C)
φ
L p (R+0 , C)
3.6
φ
Tt
L p (R+0 , C).
p
The hypothesis on B let us conclude that {τt }t≥0 is distributionally chaotic on Lρ R0 , C,
see 7, Th. 2.3. Therefore, the conclusion is obtained since distributional chaos is preserved
under conjugacy 6, Th. 2.
Remark 3.2. The previous result can be compared with the characterizations of hypercyclicity
p
and Devaney chaos for the translation C0 -semigroup on the spaces Lρ R0 , C, 1 ≤ p < ∞: The
p
translation C0 -semigroup {τt }t≥0 is hypercyclic on Lρ R0 , C if and only if, lim infx → ∞ ρx ∞
0 12, and {τt }t≥0 is Devaney chaotic on it if, and only if, 0 ρxdx < ∞ 16, 17. Using
conjugacy, these results can be transferred to the C0 -semigroup {Tt }t≥0 1, Ex. 7.5.2.
On the one hand, if hx is constant and equal to 1, then we have that {Tt }t≥0
is Devaney chaotic and distributionally chaotic on Lp R0 , C. On the other hand, taking
B 0, 2 ∪ n∈N n2 1, n 12 , hx 1 if x ∈ B and hx −1 elsewhere, we have
DensB 1 and B ρxdx < ∞. Therefore {Tt }t≥0 is distributionally chaotic on Lp R0 , C. It
2
is also hypercyclic since ρn2 e−n −2n4p for n ≥ 2, which
yields that lim infx → ∞ ρx 0
14, Th. 2.2. However, it cannot be Devaney chaotic since R ρxdx ∞.
0
To sum up, we have an example of a C0 -semigroup that is hypercyclic, distributionally
chaotic, but it is not Devaney chaotic. This example can be compared with the example
provided in 7, Ex. 2. of a distributionally chaotic translation C0 -semigroup that is neither
hypercyclic nor chaotic.
Now, let us consider another example of a C0 -semigroup whose dynamical behavior
was already discussed in 14: Let ρ : 0, 1 → R be a continuous function such that there
exist constants M ≥ 1, ω ∈ R, and γ < 0 such that
ρx ≤ Meωt ρ eγt x ,
∀x ∈ 0, 1, t > 0.
3.7
p
With such a function ρ, we can consider the spaces Lρ 0, 1, C, for 1 ≤ p < ∞. The family of
operators {St }t≥0 with St fx feγt x, t ≥ 0 defines a C0 -semigroup on them 14.
p
Theorem 3.3. If γ < 0, then the C0 -semigroup {St }t≥0 is distributionally chaotic on Lρ 0, 1, C,
1 ≤ p < ∞.
Proof. Let us apply Theorem 2.3. Take X0 {f ∈ C0, 1, C : f0 0}. This set is dense
p
in Lρ 0, 1, C and, clearly, limt → ∞ St f 0 for every f ∈ X0 , which fulfills condition i in
Theorem 2.3.
∞
Let us prove that 0 St −1
p,ρ dt is finite: Fix t > 0 and a continuous function g on 0, 1
with gp,ρ 1, for instance gx 1/ρx1/p .
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Abstract and Applied Analysis
There is some t0 > 0 such that for t > t0 , we have t > t0 , define
eγt
0
1/p
ρxdx
≤ t−2 /g∞ . For these
g e−γt x , if 0 ≤ x ≤ eγt ,
gt x 0,
elsewhere.
Since gt p,ρ ≤ t−2 and St gt g, then St p,ρ ≥ t2 for t ≥ t0 . So that
which yields the conclusion.
3.8
∞
t0
St −1
p,ρ dt is convergent,
Remark 3.4. The assumption γ < 0 forces w > 0: If not, take any x ∈ 0, 1. Taking limits
when t → ∞ in the inequality ρx/ρeγt x ≤ Meωt we have ρx/ρ0 ≤ 0, which is a
contradiction because ρ is a positive continuous function.
Remark 3.5. An alternative proof provided by an anonymous referee is the following: If ρ :
0, 1 → R is a continuous weight function which is admissible in the sense of 3.7, then
ψ : R0 → R0 defined as ψx : ρeγx eγx is an admissible weight function in the sense of 12,
p
p
Def. 4.1. Therefore, taking φ : Lψ R0 , C → Lρ 0, 1, C defined as φfx : flogx/γ,
we have the following commutative diagram:
p
Lψ (R+0 , C)
τt
3.9
φ
φ
p
Lρ ([0, 1], C)
p
Lψ (R+0 , C)
St
p
Lρ ([0, 1], C).
∞
If γ < 0, then 0 ψxdx < ∞. So that, by conjugacy, {St }t≥0 is hypercyclic, Devaney and
distributionally chaotic, see Remark 3.2.
We return to the initial value problem stated in 3.2. Consider the case when I 0, 1,
ζx : γx, γ < 0, and h ∈ C0, 1, C. Under these hypotheses, the C0 -semigroup {Tt }t≥0
defined as
t
γt−r
Tt fx exp
h e
x dr f eγt x
for t ≥ 0, x ∈ 0, 1,
3.10
0
gives the solution C0 -semigroup to 3.2 on Lp 0, 1, C, 1 ≤ p < ∞ 14, Th. 3.4. The
particular case when γ −1 and hx −1/2 was studied using the Wiener measure in
15.
Theorem 3.6. If γ < 0 and min{hx : x ∈ 0, 1} > γ/p, then the C0 -semigroup {Tt }t≥0 defined
in 3.10 is distributionally chaotic on Lp 0, 1, C, 1 ≤ p < ∞.
Proof. We apply again Theorem 2.3: Condition i holds in the same way as in the proof of
Theorem 3.3 taking X0 {f ∈ C0, 1, C : f0 0}.
Abstract and Applied Analysis
7
In order to verify condition ii, let α ∈ R be such that min{hx : x ∈ 0, 1} > α >
γ/p. For every t > 0, we define ft as a function with ||ft ||p 1 and suppft ⊂ 0, eγt . Using
it, we have the following estimations for ||Tt ||p :
1
eγt
1/p
1/p
γt p
p
αt
α−γ/pt
ft e x dx
ft y
e
dy
eα−γ/pt
Tt ≥ Tt ft ≥ e
p
p
0
0
3.11
so that
∞
0
Tt −1
p dt is finite, which yields the conclusion.
Under the hypothesis of the last theorem, Takeo proved that {Tt }t≥0 is Devaney chaotic
by applying the Desch-Schappacher-Webb Criterion 14. Independently, Brzeźniak and
Dawidowicz also proved that {Tt }t≥0 is Devaney chaotic when γ −1 and hx λ ∈ R
with λ > −1/p, that is known as the von Foerster-Lasota equation 18, Theorems 8.3 and 8.4.
Furthermore, they also showed that for λ ≤ −1/p the orbits of all elements tend to 0, which
makes chaos disappear. Therefore, we can affirm that Devaney chaos coincides exactly with
distributional chaos for the same values of λ. As we will see later, this is due to the fact that
Devaney chaos can be obtained here from the Desch-Schappacher-Webb Criterion. This can
be easily seen if we reformulate Theorem 2.3 in terms of the infinitesimal generator of the
C0 -semigroup. The following result is a continuous version of 4, Cor. 31.
Theorem 3.7. Let X be a complex Banach space and let T be a C0 -semigroup in LX with
infinitesimal generator A, DA. If the following conditions hold:
i there is a dense subset X0 ⊂ X with limt → ∞ Tt x 0, for each x ∈ X0 , and
ii there is some λ ∈ σp A with λ > 0,
then T has a dense distributionally irregular manifold. In particular, T is distributionally chaotic.
Proof. Fix t > 0. On the one hand, if condition i holds, then we have limn → ∞ Ttn x 0 for
every x ∈ X0 . On the other hand, by the point spectral mapping theorem for C0 -semigroups,
since λ ∈ σp A, then eλt ∈ σp Tt . Therefore rTt ≥ |eλt | > 1 and, by 4, Cor. 31, Tt admits
a dense distributionally irregular manifold. By 8, Rem. 2, this is equivalent to say that T
admits a dense distributionally irregular manifold. Furthermore, T is distributionally chaotic
8, Prop. 2.
Remark 3.8. Clearly, the conditions in Theorem 3.7 hold whenever the Desch-SchappacherWebb Criterion can be applied. Therefore, among others, the following C0 -semigroups
that are known to be Devaney chaotic are also distributionally chaotic and have a dense
distributionally irregular manifold: 19, 20, Th. 3.1, 12, Ex. 4.12, 21, 22, Th. 1, and
23, Th. 2.1 and Th. 2.3. See also 1, Ch. 7 for an improved version of the proof of these last
two examples.
8
Abstract and Applied Analysis
Finally, Brzeźniak and Dawidowicz also studied in 18 Devaney chaos for the case
γ −1 and hx λ ∈ R in certain subspaces of Hölder continuous functions on 0, 1. For
α ∈0, 1, 0 < r ≤ 1, we define the space Crα 0, 1 of functions f : 0, 1 → R such that
f :
α,r
fx − f y sup
< ∞.
x − yα
x,y∈0,1
3.12
0<|x−y|<r
For α ∈0, 1, let us consider Vα 0, 1 the space of functions
f ∈ C1α 0, 1 : lim f α,r 0, f0 0 .
r →0
3.13
In 18, it is shown that Vα 0, 1 is a separable Banach space endowed with the norm
||f||α,1 . Furthermore, following a constructive approach, it is proved that if γ −1 and
hx λ > α, then {Tt }t≥0 is Devaney chaotic there. We will prove that in this case {Tt }t≥0
is also distributionally chaotic.
Theorem 3.9. If γ −1 and hx λ > α, then the C0 -semigroup {Tt }t≥0 defined in 3.10 is
distributionally chaotic on Vα 0, 1, α ∈0, 1.
Proof. We will apply Theorem 2.3 again. Since {Tt }t≥0 is Devaney chaotic, then there is a
dense set of points with bounded orbit. Therefore {Tt }t≥0 is weakly mixing 1, Th. 7.23, and
any non-trivial operator Tt is weakly mixing, too 11, Th. 2.4. Fix t > 0. By 9, Th. 2.3, Tt
satisfies the Hypercyclicity Criterion. So that, there is a dense set X0 ⊂ Vα 0, 1 such that
limn → ∞ Ttn x 0 for all x ∈ X0 . Using the local equicontinuity of {Tt }t≥0 , we have limt → ∞ Tt x 0 for every x ∈ X0 and condition i holds.
In order to verify condition ii, take 0 < ε < λ − α/2 such that α ε < 1. Let us define
fε x xαε , 0 ≤ x ≤ 1. Since |xαε − yαε | ≤ |x − yαε | for all x, y ∈ 0, 1, then we can
easily see that ||fε ||α,1 1 and fε ∈ Vα 0, 1. We also get that ||Tt fε ||α,1 eλ−α−εt and hence
∞
∞
dt/Tt α,1 ≤ 0 dt/Tt fε α,1 < ∞.
0
4. Discussion and Conclusions
Consider the initial value problem of 3.2 on L1 R0 , C with ζx 1 and hx kxk−1 /1 xk . Here, the solution C0 -semigroup {Tt }t≥0 is defined as
Tt fx 1 x tk
fx t,
1 xk
x, t ≥ 0.
4.1
The C0 -semigroup {Tt }t≥0 defined in 4.1 is distributionally chaotic on L1 R0 , C by
Theorem 3.1. The hypercyclicity of this C0 -semigroup for k 2 was obtained by El Mourchid
in 24 and the Devaney chaos by Grosse-Erdmann and Peris in 1, Prop. 7.34. In this case,
the point spectrum of the infinitesimal generator is the closed left half plane. This inhibits
the Desch-Schappacher-Webb Criterion to be applied in the way it has been formulated.
Nevertheless, El Mourchid observed that the hypercyclic behavior of this C0 -semigroup is
Abstract and Applied Analysis
9
essentially due to the imaginary eigenvalues of its infinitesimal generator 24, see also 1, Ex.
7.5.1. In fact, the Desch-Schappacher-Webb Criterion can be strengthened and reformulated
as follows.
Theorem 4.1 see 24, Th. 2.1 and 1, Th. 7.31. Let X be a complex separable Banach space,
and let T be a C0 -semigroup on X with infinitesimal generator A, DA. If there are a < b and
continuous functions fj : a, b → X, j ∈ J, with
1 fj s ∈ keris I − A for every s ∈ a, b, j ∈ J, and
2 span{fj s; s ∈ a, b, j ∈ J} is dense in X,
then the semigroup T is Devaney chaotic.
To sum up, we have seen that even when we apply this stronger version of the DeschSchappacher-Webb Criterion for the C0 -semigroup in 4.1, asking only for an abundance
of eigenvalues of real part equal to zero, then we can also prove that there is a dense
distributionally irregular manifold. Therefore, we can pose the following problem.
Problem 1. Do the hypothesis in Theorem 4.1 imply the existence of a dense distributionally
irregular manifold for T? If not, is there at least a distributionally irregular vector for T?
By the equivalence between a C0 -semigroup with a distributionally irregular vector
and a distributionally chaotic C0 -semigroup, 8, Th. 3.4, the former problem can also be
presented as follows.
Problem 2. Do the hypothesis in Theorem 4.1 imply that T is distributionally chaotic?
These questions could have a positive answer, but it is still unknown whether Devaney
chaos implies distributional chaos on C0 -semigroups.
Problem 3. Are there examples of Devaney chaotic C0 -semigroups which are not distributionally chaotic?
A C0 -semigroup is said to be frequently hypercyclic if there exists some x ∈ X such
that for every nonempty open set U ⊂ X, the set Ux : {s ≥ 0 : Ts x ∈ U} has positive
lower density, that is lim inft → ∞ 1/tμUx ∩ 0, t is positive. In 25, Mangino and Peris
observed that with the same arguments used in 12, 24, one can show that the DeschSchappacher-Webb Criterion implies frequent hypercyclicity. They also provide the Frequent
Hypercyclicity Criterion for C0 -semigroups 25, Th. 2.2. So that, one can raise the following
question.
Problem 4. Do the hypothesis of the Frequent Hypercyclicity Criterion for C0 -semigroups
imply distributional chaos?
The hypothesis in Theorem 4.1 also yields the mixing property for the C0 -semigroup
T, see 1. We recall that a C0 -semigroup is topologically mixing if for any pair of nonempty
∅ for all t ≥ t0 . Clearly, topological
open sets U, V ⊂ X there is some t0 > 0 such that Tt U∩V /
mixing implies transitivity i.e., hypercyclicity, but it is strictly stronger than it. Topologically
p
mixing translation C0 -semigroups on the weighted Lρ -spaces considered in this paper are
characterized by the condition limt → ∞ ρt 0 26, Th. 4.3.
10
Abstract and Applied Analysis
On the one hand, the aforementioned example of Peris and Barrachina 7, Ex. 2.7
provides an example of a distributionally chaotic C0 -semigroup that it is not topologically
mixing. On the other hand, in 27, there is an example of a backward shift operator on
a weighted sequence space p v, 1 ≤ p < ∞, that is topologically mixing but it is not
distributionally chaotic. This operator will provide us an analogous counterexample in the
frame of C0 -semigroups. We thank A. Peris for this counterexample.
Example 4.2. Consider the sequence nk k defined as nk k!3 , k ∈ N, and define the function
ρ : R0 → R0 as ρt 1 if 0 ≤ t. This function is an admissible weight in the sense of
12, Def. 4.1 and makes the translation semigroup {τt }t≥0 to be a C0 -semigroup. On the one
hand, since limt → ∞ ρt 0, then the translation C0 -semigroup is topologically mixing. On
the other hand, if the translation C0 -semigroup was distributionally chaotic, by 7, Th. 2.10,
the backward shift operator, defined as Bx1 , x2 , . . . x2 , x3 , . . ., would be distributionally
chaotic on the space 1 v : {xn n : j∈N |xj |vj < ∞} with vn n ρnn , which is a
contradiction as it is indicated in 27.
Acknowledgments
This work is supported in part by MEC and FEDER, Project MTM2010-14909, by Generalitat
Valenciana, Project GV/2010/091, and by Universitat Politècnica de València, Project PAID06-09-2932. X. Barrachina also wants to acknowledge the support of the Grant FPI-UPV 200904 from Programa de Ayudas de Investigación y Desarrollo de la Universitat Politècnica
de València. The authors also thank the referees for helpful comments that improved the
presentation of the paper.
References
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2 F. Bayart and É. Matheron, Dynamics of Linear Operators, vol. 179, Cambridge University Press,
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